@andersen_800: "My book says that "we will say that the expectation is well-defined if
\integral_-oo^oo |x|f(x)dx < oo". Then they give an example of a not 
well-defined PDF:
f(x)=c/(1+x^2) for some normalization constant c. The |x|f(x) can bbe 
approximated by c/|x| for large |x|. Using the fact that
\integral_1^oo 1/x dx = oo, one can show that
\integral_-oo^oo |x|f(x)dx = oo.
Thus, E[X] is left undefined, despite the symmetry of the PDF around zero.
Now to my question. The example just showed it to be undefined for the 
case which integrates |x|f(x), not for the proper definition of E[X] 
which integrates x f(x). Thus, E[X] might not be undefined, only their 
litmus test |x|f(x). Am I missing something?"
    @kirakun: "<quote>
In real analysis, by definition a function is integrable if and only if 
both its positive part and negative part are integrable.  This is 
equivalent to saying the function integrable if and only if it is 
absolutely integrable.
-kira"
    @ullrich: "On Wed, 08 Aug 2007 11:24:49 +0200, Andersen
<quote>
Yes, you;re missing the definitions at the bottom of it all.
(Which may or may not be your fault...)
Probability is based on measure theory, and in measure theory
a function is "integrable" if and only if its absolute value
has a finite integral.
There are reasons for that - if we allowed RV's like the one
above to have expectations based on symmetry even though
they're not absolutely integrable the theorems of
probability theory wouldn't work.
************************
David C. Ullrich"
    @andersen_800: "<quote>
Not completely my fault. This book (Bersekas et al) does not go through 
measure theory. I do have Williams (which I guess is the standard 
reference), but I find it very difficult to read. It feels like I should 
first start with a proper real analysis book (like Kolmogorov?), because 
I only studied calculus as undergrad. Williams seem to have picked the 
smallest possible subset of real analysis required to do probability, 
and things are under-explained IMHO. Hence, I really don't have the 
formal treatment of probability theory through measure theory. Any 
analysis textbook to recoomend (which is not too difficult for a non 
math student).
I see, I did not know this.
I see. That is understandable.
Thanks."
    @sjherschko: "<quote>
The book stated the author's definition and showed an example which 
satisfied it.  But I understand your question.  By the authors' 
definition,  EX  is not well-defined in two cases:  where  
integral(-infty..infty, f)  is infinity (or -infinty),  and where the 
improper integral diverges because integral(a..b, f) can take both 
arbitrarily large and arbitrarily small values for different choices of  
a  and  b.  I think you agree that  EX  is truly not well-defined in the 
latter case.  Personally, I would take an infinite expectation to be 
"defined," but that is a matter of convention.
However, the distinction between the two cases is important.  Let  X1, 
X2, X3,...  be a sequence of independent, identically distributed random 
variables.
If  EX1 = infty,  then, with probability 1,  sum(i=1..n, X_i)/n  grows 
arbitrarily large as n approaches infinity.
If  EX1 is undefined and not infinity, then the same is true of the 
limit of  sum(i=1..n, X_i)/n  (with probability 1).
Stephen J. Herschkorn                        <email_address>
Math Tutor on the Internet and in Central New Jersey and Manhattan"
    @sjherschko: "<quote>
 From the looks of it, you are not a math major (as you just stated), 
and you are strudyig probability for the first time.  In this case, I 
strongly recommend that you do *not*  get bogged down in the 
measure-theoretic underpinnings of the subject.  In a first course, your 
goal should be to become comfortable with the calculus of applied 
probability, learning how to apply this calculus to problems 
(particularly ones in your field), and developing a certain amount of 
probabilistic inuition.  Your calculus background should suffice.  A 
measure-theoretic approach will only obcure the basic ideas.  If you 
really want to learn those, do so *after* having developed the applied 
knowledge.
If you find your current text wanting, try looking at other applied 
first course texts.  My favorite it  Ross SM, A First Course in 
Probability.  You could probably purchase a used old edition rather 
cheaply.  You might take also take a look at Drake AW, Fundamentals of 
Applied Probability Theory, though you might find too extreme the 
minimization of the formal mathematics in his presentation.  Note that 
you don't have to buy other sources; you can always go to the library.
I would say that probability, which has existed for centuries, was 
*formalized* through measure theory.  Most of it got on quite well 
before then.
By the way, in measure theory, the integral of a measurable function is 
still *defined* in certain cases where the function is not integrable 
(though the terminology  may seem paradoxical).  For a function is 
allowed to have its integral take on the value of infinity.
The upshot, I think:  If  EX is defined but infinite,  you can still 
work with the expectation.  If you can't even speak of any finite or 
infinte value for the expected value, then EX is not defined.  So,  for 
example,
if P{X = n} = c/n^2  for  n = 1, 2, 3,...  EX exists and is infinite, 
thought your book says EX is not "well-defined."  If  P{X = n} =  c/(2 
n^2)  for  n any nonzero integer (including the negative ones),  then  
EX is not even deifined.  I think the authors should have used the more 
common term, "integrable," rather than "well-defined."
Stephen J. Herschkorn                        <email_address>
Math Tutor on the Internet and in Central New Jersey and Manhattan"
    @sjherschko: "<quote>
I make the assumption that the author maintains the same convention for 
discrete and continuous random variables.
As discussed, the expected value of such a random variable is 
undefined.   However, that observation is not the point of my comment.  
Here is the OP's first post.
The OP infers (reasonably, in my opinion) that this definition implies 
that the expectation of a random variable with density  x |-> 1_{x>1} / 
x^2  is not "well-defined."  My point is that such a convention is 
misleading.  In this case,  EX is defined and equals (positive) 
infinity.  One frequently encounters such random variables in 
applications and theory.
Now that I reexamine the original quote of Bertsekas, I do see that he 
does say "if", not "if and only if," so I supposethe OP and I could be a 
misreading.  However, had the author really only meant "if," that would 
have been a very silly way to present the terminology.
Stephen J. Herschkorn                        <email_address>
Math Tutor on the Internet and in Central New Jersey and Manhattan"
    @andersen_800: "<quote>
Thanks for this information. It is very helpful. I am happy with the 
book I have. I think if I ever want to take the measure theoretic 
approach, I will first study real analysis before going off to measure 
theory.
Just as Leibniz did well without limits I guess. Infinitesimals were 
sufficient.
Ok.
I see your point. It is just like lim inf and lim sup which can take 
infinite values, and always have a value, unlike inf and sup which might 
be undefined."
    @andersen_800: "<quote>
It really only had one implication, I actually also reacted on that. So 
it is "if" and not "iff"."
    @sjherschko: "<quote>
But don't you see why it is kind of silly to read that as literally an 
"if" only?  With that literal interpretation, if  
integral(x=-infty..infty, |x| f(x)) = infty,  we don't know if  EX  is 
"well-defined" or not.  So the fact that a Cauchy random variable (the 
example presented) does not have a well-defined expectation does not 
follow from the first statement.
BTW, I see your first post was a bit imprecise:  The Cauchy *PDF* is 
quite well-defined; the *expectation* is undefined.
I was unaware that Bertsekas had written a probability text, since I 
have not been in the academy since before its publication.  I was 
surprised to learn of it, since Bertsekas is known primarily as a math 
programming guy (or at least he used to be).  It looks like he has taken 
over the course Al Drake (may he rest in peace) smartin@example.com. for 
years.  When Drake taught it, he used his own text book (as many 
professors do).
Stephen J. Herschkorn                        <email_address>
Math Tutor on the Internet and in Central New Jersey and Manhattan"
    @andersen_800: "<quote>
Yes you are right.
Did not know about Al Drake. Now I found the whole PDF of his book on 
the net. Looks good. Bertsekas seems better though (Drake has 
statistics, but not much on markov processes, Bertsekas is the other way 
around)."
    @sjherschko: "<quote>
Are you looking at the book I mean?
Drake AW.  Fundamentals of Applied Probability Theory.  New York:  
McGraw-Hill, 1967.  ISBN 07-017815-1
The entire fifth chapter is on discrete- and continuous-time Markov 
processes.  Chapter 7 presents only a brief introduction to inferential 
statistics.  It looks like it's out of print.  I think it is a great 
book for learning the *intuition* of probability, and it is the epitome 
of the treatment of probability as a subject that is not merely a subset 
of mathematics.  My only criticism is that Drake seems to go out of his 
way to avoid "high-fallutin'" (as he would have said) mathematical 
notation and rigor.  For someone like me, that makes the book harder to 
understand than Ross.
Al Drake was a wonderful, very funny (as in comical), and beloved 
professor, by the way.  The humor shows in many of the exercises in the 
book.  For example, one starts, "There are 240 students in a literature 
class ('Proust, Joyce, Kafka, and Mickey Spillane')."  And Oscar and his 
dog are (in?)famous amongst the alumni of his course.  
<url> presents a 
nice memorial.
Stephen J. Herschkorn                        <email_address>
Math Tutor on the Internet and in Central New Jersey and Manhattan"
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